Homogeneous Carter diagrams have similar signed enhanced Dynkin diagrams

Let $\{\widetilde\Gamma,\Gamma\}$ be a homogeneous pair of Carter diagrams. Associate signed enhanced Dynkin diagrams $\Delta(\widetilde\Gamma)$ and $\Delta(\Gamma)$ to them. Homogeneous-pair conjecture. The signed enhanced Dynkin diagrams associated with $\widetilde\Gamma$ and $\Gamma$ coincide up to similarities:

$$\Delta(\widetilde\Gamma)=\Delta(\Gamma).$$

The paper presents this as a further conjecture concerning the relationship between enhanced Dynkin diagrams and Carter diagrams; the supplied excerpt gives no evidence that it has been resolved.